Lots of refactoring towards partially-ordered ring R (#109)

Numbers/Rationals/Definition.agda

@@ -1,4 +1,4 @@
-{-# OPTIONS --safe --warning=error #-}
+{-# OPTIONS --safe --warning=error --without-K #-}
 
 open import LogicalFormulae
 open import Numbers.Naturals.Naturals
@@ -10,7 +10,6 @@ open import Rings.Definition
 open import Rings.Orders.Total.Definition
 open import Rings.Orders.Partial.Definition
 open import Fields.Fields
-open import Numbers.Primes.PrimeNumbers
 open import Setoids.Setoids
 open import Setoids.Orders
 open import Functions
@@ -28,6 +27,9 @@ open import Fields.FieldOfFractions.Order ℤIntDom ℤOrderedRing
 ℚ : Set
 ℚ = fieldOfFractionsSet
 
+ℚSetoid : Setoid ℚ
+ℚSetoid = fieldOfFractionsSetoid
+
 _+Q_ : ℚ → ℚ → ℚ
 a +Q b = fieldOfFractionsPlus a b
 
@@ -43,6 +45,9 @@ a *Q b = fieldOfFractionsTimes a b
 injectionQ : ℤ → ℚ
 injectionQ z = z ,, (nonneg 1 , λ ())
 
+injectionQInjective : Injection injectionQ
+injectionQInjective pr = equalityLeft pr
+
 ℚField : Field ℚRing
 ℚField = fieldOfFractions
 

Numbers/Reals/Definition.agda

@@ -1,9 +1,11 @@
-{-# OPTIONS --warning=error --safe --guardedness #-}
+{-# OPTIONS --warning=error --safe --guardedness --without-K #-}
 
 open import Setoids.Orders
+open import Setoids.Setoids
 open import LogicalFormulae
 open import Rings.Definition
 open import Numbers.Rationals.Definition
+open import Functions
 
 module Numbers.Reals.Definition where
 
@@ -23,12 +25,20 @@ _+R_ = _+C_
 _*R_ : ℝ → ℝ → ℝ
 _*R_ = _*C_
 
+ℝSetoid = cauchyCompletionSetoid
+
+_=R_ : ℝ → ℝ → Set
+a =R b = Setoid._∼_ cauchyCompletionSetoid a b
+
 ℝRing : Ring cauchyCompletionSetoid _+R_ _*R_
 ℝRing = CRing
 
 injectionR : ℚ → ℝ
 injectionR = injection
 
+injectionRInjective : Injection injectionR
+injectionRInjective = CInjection'
+
 0R : ℝ
 0R = injection 0Q